Theorem lincval 42198

Description: The value of a linear combination. (Contributed by AV, 30-Mar-2019.)

Assertion

Ref	Expression
lincval	|- ( ( M e. X /\ S e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ V e. ~P ( Base ` M ) ) -> ( S ( linC ` M ) V ) = ( M gsumg ( x e. V |-> ( ( S ` x ) ( .s ` M ) x ) ) ) )

Distinct variable groups:   x, M   x, S   x, V

Allowed substitution hint:   X( x)

Proof of Theorem lincval

Dummy variables s v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lincop 42197	. . . 4 |- ( M e. X -> ( linC ` M ) = ( s e. ( ( Base ` (Scalar ` M ) ) ^m v ) , v e. ~P ( Base ` M ) |-> ( M gsumg ( x e. v |-> ( ( s ` x ) ( .s ` M ) x ) ) ) ) )
2	1	3ad2ant1 1082	. . 3 |- ( ( M e. X /\ S e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ V e. ~P ( Base ` M ) ) -> ( linC ` M ) = ( s e. ( ( Base ` (Scalar ` M ) ) ^m v ) , v e. ~P ( Base ` M ) |-> ( M gsumg ( x e. v |-> ( ( s ` x ) ( .s ` M ) x ) ) ) ) )
3	2	oveqd 6667	. 2 |- ( ( M e. X /\ S e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ V e. ~P ( Base ` M ) ) -> ( S ( linC ` M ) V ) = ( S ( s e. ( ( Base ` (Scalar ` M ) ) ^m v ) , v e. ~P ( Base ` M ) |-> ( M gsumg ( x e. v |-> ( ( s ` x ) ( .s ` M ) x ) ) ) ) V ) )
4		simp2 1062	. . 3 |- ( ( M e. X /\ S e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ V e. ~P ( Base ` M ) ) -> S e. ( ( Base ` (Scalar ` M ) ) ^m V ) )
5		simp3 1063	. . 3 |- ( ( M e. X /\ S e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ V e. ~P ( Base ` M ) ) -> V e. ~P ( Base ` M ) )
6		ovexd 6680	. . 3 |- ( ( M e. X /\ S e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ V e. ~P ( Base ` M ) ) -> ( M gsumg ( x e. V |-> ( ( S ` x ) ( .s ` M ) x ) ) ) e. _V )
7		simpr 477	. . . . . 6 |- ( ( s = S /\ v = V ) -> v = V )
8		fveq1 6190	. . . . . . . 8 |- ( s = S -> ( s ` x ) = ( S ` x ) )
9	8	oveq1d 6665	. . . . . . 7 |- ( s = S -> ( ( s ` x ) ( .s ` M ) x ) = ( ( S ` x ) ( .s ` M ) x ) )
10	9	adantr 481	. . . . . 6 |- ( ( s = S /\ v = V ) -> ( ( s ` x ) ( .s ` M ) x ) = ( ( S ` x ) ( .s ` M ) x ) )
11	7, 10	mpteq12dv 4733	. . . . 5 |- ( ( s = S /\ v = V ) -> ( x e. v |-> ( ( s ` x ) ( .s ` M ) x ) ) = ( x e. V |-> ( ( S ` x ) ( .s ` M ) x ) ) )
12	11	oveq2d 6666	. . . 4 |- ( ( s = S /\ v = V ) -> ( M gsumg ( x e. v |-> ( ( s ` x ) ( .s ` M ) x ) ) ) = ( M gsumg ( x e. V |-> ( ( S ` x ) ( .s ` M ) x ) ) ) )
13		oveq2 6658	. . . 4 |- ( v = V -> ( ( Base ` (Scalar ` M ) ) ^m v ) = ( ( Base ` (Scalar ` M ) ) ^m V ) )
14		eqid 2622	. . . 4 |- ( s e. ( ( Base ` (Scalar ` M ) ) ^m v ) , v e. ~P ( Base ` M ) |-> ( M gsumg ( x e. v |-> ( ( s ` x ) ( .s ` M ) x ) ) ) ) = ( s e. ( ( Base ` (Scalar ` M ) ) ^m v ) , v e. ~P ( Base ` M ) |-> ( M gsumg ( x e. v |-> ( ( s ` x ) ( .s ` M ) x ) ) ) )
15	12, 13, 14	ovmpt2x2 42119	. . 3 |- ( ( S e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ V e. ~P ( Base ` M ) /\ ( M gsumg ( x e. V |-> ( ( S ` x ) ( .s ` M ) x ) ) ) e. _V ) -> ( S ( s e. ( ( Base ` (Scalar ` M ) ) ^m v ) , v e. ~P ( Base ` M ) |-> ( M gsumg ( x e. v |-> ( ( s ` x ) ( .s ` M ) x ) ) ) ) V ) = ( M gsumg ( x e. V |-> ( ( S ` x ) ( .s ` M ) x ) ) ) )
16	4, 5, 6, 15	syl3anc 1326	. 2 |- ( ( M e. X /\ S e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ V e. ~P ( Base ` M ) ) -> ( S ( s e. ( ( Base ` (Scalar ` M ) ) ^m v ) , v e. ~P ( Base ` M ) |-> ( M gsumg ( x e. v |-> ( ( s ` x ) ( .s ` M ) x ) ) ) ) V ) = ( M gsumg ( x e. V |-> ( ( S ` x ) ( .s ` M ) x ) ) ) )
17	3, 16	eqtrd 2656	1 |- ( ( M e. X /\ S e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ V e. ~P ( Base ` M ) ) -> ( S ( linC ` M ) V ) = ( M gsumg ( x e. V |-> ( ( S ` x ) ( .s ` M ) x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   ~Pcpw 4158   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^m cmap 7857   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   gsum_g cgsu 16101   linC clinc 42193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-linc 42195

This theorem is referenced by:  lincfsuppcl  42202  linccl  42203  lincval0  42204  lincvalsng  42205  lincvalpr  42207  lincvalsc0  42210  linc0scn0  42212  lincdifsn  42213  linc1  42214  lincellss  42215  lincsum  42218  lincscm  42219  lindslinindimp2lem4  42250  lindslinindsimp2lem5  42251  snlindsntor  42260  lincresunit3lem2  42269  lincresunit3  42270  zlmodzxzldeplem3  42291